Outline

• Motivation
• Working Example
• Model
• Visualizations
  • Static
  • Animation
  • Interactivity with plotly

Motivating Example

Childhood vaccination - major public health concern

• Vaccination animation

Questions

Interactions in animation aren’t realistic - people don’t behave like bacteria in petri dishes!

• How do communities affect vaccination rate?
• How do communities affect spread of disease from outside source?
• What factors lead to unvaccinated children?
• Can communities be identified and targeted for immunization programs?

Can network modeling answer these?

Working Example

Another public health concern - underage drinking & smoking

How do friendships influence drinking and smoking behavior in teenagers?

Steglich, Snijders & Pearson (2010) “Dynamic Networks and Behavior: Searating Selection from Influence”

Data

From “Teenage Friends and Lifestyle Study” (Pearson & Mitchell (2000); Pearson & West (2003))

• Glasgow, Scotland, UK, Feb 1995 - Jan 1997
• Panel data - three waves - aged 12-13 to 14-15
• Each student lists up to 6 school friends
• Smoking & drinking behavior, other lifestyle variables collected
• 160 students in total, only 50 in example data

Model

• Want: to determine how friendships change risky behavior and vice versa over time
• Solution: stochastic actor-oriented models (SAOMs) for networks (Snijders 1996)
  • Stochastic because they model network change over time
  • Actor-oriented because actor-level covariates are incorporated into model parameters XX EXAMPLE XX

Underlying Mechanisms?

Network changes

• Change doesn’t happen all at once, but network observations are discrete
• Changes are dependent, with a highly complex structure
• How to model process?
  • Continuous-time Markov chains (CTMCs)
  • Condition on wave 1
  • Choose one actor (call it \(i\)) to (potentially) make a change
  • Optimize some utility function by changing ties - choose “best” friend to have or “worst” friend to cut loose

Rate Function

Rate at which actors are selected to act

• Simple rate parameter, \(\alpha\), for all actors
• Waiting time for one actor to act is distributed \(Exp(\alpha^{-1})\)
• Waiting time for any actor to act is distributed \(Exp((n\alpha)^{-1})\)
• Probability that actor \(i\) will be the next one to act is \(\frac{1}{n}\)

Transition Steps

Objective Function

When actor \(i\) can change, it tries to maximize its objective function:

\[f_i(x, \mathbf{z}, \boldsymbol{\beta}) = \sum_k \beta_k s_{ik}(x, \mathbf{z})\]

• \(x\) is the network state
• \(\mathbf{z}\) is a matrix of actor-level covariates
• \(s_{ik}(x, \mathbf{z})\) are network & covariate statistics
• \(\beta_k\) are the parameters to be fit in the model
• There can be any number of \(\beta_k\) added to the model (problematic)

Transition Probability

The probability that actor \(i\) will change the tie to actor \(j\) is:

\[p_{ij} = \frac{\exp\left\{f_i(x(i\leadsto j), \mathbf{z}, \boldsymbol{\beta})\right\}}{\sum_h \exp\left\{f_i(x(i\leadsto h), \mathbf{z}, \boldsymbol{\beta})\right\}}\] - \(x(i\leadsto j)\) is the network identical to the current state, \(x\), except for \(x_{ij}\), which becomes \(1-x_{ij}\)

Transition Probability Heatmap

(Some) Possible Model Parameters

(Some) Possible Model Parameters

Structural Effect	Interpretation	Visualization
outdegree*	Popularity
reciprocity*	Reciprocated relationships
transitive triplets	Your friend becomes my friend

Covariate Effect	Interpretation	Visualization
covariate-alter	Effect of my friend’s behavior on friendship
covariate-ego	Effect of my behavior on friendship
same covariate	Birds of a feather flock together

Small SAOMs Example

• Small subset of the 50 friends data
• Fit three models for network evolution
• M1: “straw man” model that only has 2 parameters in obective function
• M2: M1 with one significant covariate effect added (drinking behavior encourages transitive ties)
• M3: M1 with one significant structural effect added (discourages transitive ties, encourages social distance)

Static Visualizations

• Collections of models & corresponding parameter estimates
• “Heatmap” of probabilities

Distributions of Fitted Model Effects

Correlation of Estimates

Animation

• What does the underlying CTMC actually look like?

GIF of Awesomeness

Interactivity with plotly

• geomnet + ggplotly()

Future Work

• Shiny app for more interactivity
• Simulation studies with more visualization

References

Pearson, M. and Michell, L. 2000. “Smoke Rings: Social Network Analysis of Friendship Groups, Smoking, and Drug-Taking.” Drugs: Education, Prevention and Policy 7(1):21-37.

Pearson, M. and West, P. 2003. “Drifting Smoke Rings: Social Network Analysis and Markov Processes in a Longitudinal Study of Friendship Groups and Risk-Taking.” Connections 25(2):59-76.

Steglich, C., Snijders, T.A.B, and Pearson, M. 2010. “Dynamic Networks and Behavior: Separating Selection from Influence.” Sociological Methodology. 40(1):329-393.

Snijders, T.A.B. 1996. “Stochastic actor-oriented models for network change.” Journal of Mathematical Sociology 21:149-172.